# Stiefel–Whitney classes in the spirit of Chern-Weil

Chern-Weil theory gives characteristic classes (e.g. Chern class, Euler class, Pontryagin) of a vector bundle in terms of polynomials in the curvature form of an arbitrary connection. There seems to be no hope in getting Stiefel-Whitney classes from this method since Chern-Weil gives cohomology classes with real coefficients while Stiefel-Whitney classes have $\mathbb Z/2$ coefficients. Further, since any vector bundle over a curve has vanishing curvature, classes obtained by Chern-Weil can't distinguish, for example, the Mobius bundle from the trivial bundle over the circle (while Stiefel-Whitney classes do).

Nonetheless, I am wondering if there is a more general or abstract framework that allows one to define the Stiefel-Whitney classes in the spirit of Chern-Weil. For example, maybe this is done through a more abstract definition of a connection/curvature.

• Might be worth looking nLabwards, e.g., ncatlab.org/nlab/show/… or ncatlab.org/nlab/show/… or math.ntnu.no/~stacey/Mathforge/nForum/… . Jan 18, 2011 at 17:06
• I don't have an answer to this question, but I have a guess. In general, if you want to use De Rham theory to recover integral invariants of a manifold, a smart thing to do is look at distributional differential forms (I first learned of the literature on this from MO, but I can't remember where). So I would start by defining a distributional connection on a vector bundle $E$ (rather, on its frame bundle) to be a distributional $E$-valued 1-form and going from there. Jan 18, 2011 at 18:52
• You might want to take a look at "Stiefel-Whitney currents" by Harvey and Zweck [MR1731064 (2001c:53104)].
– 680
Jan 19, 2011 at 12:07
• @David and 680: thanks for the links! Jan 20, 2011 at 23:37
• @diverietti: I think the confusion is about real curves as opposed to complex curves ;-) Apr 11, 2011 at 14:41

But in certain contexts, we can get things that are analogous. For example if you take a principal $B$-bundle $Q$ over $M$ and then suppose you can have a "nice" :) central extension of your lie group $B$ by $$1 \to \underline{\mathbb{C}^*}_M \to \tilde{B} \to B \to 1,$$where $\underline{\mathbb{C}^*}_M$ is the sheaf of smooth functions into $\mathbb{C}^*$, then you can define a cohomology class in $H^1(M, \underline{B})$ by seeing how well you can lift your bundle to a $\tilde{B}$-bundle. Now by the central extension, we would have $$H^1(M, \underline{B}) \overset{\sim}{=} H^2(M , \underline{\mathbb{C}^*}_M)$$ and then by the exponential sequence you would have $$H^1(M, \underline{B}) \overset{\sim}{=} H^2(M , \underline{\mathbb{C}^*}_M) \overset{\sim}{=} H^3(M, \mathbb{Z}).$$
There exist flat manifolds (i.e., closed Riemannian manifolds with vanishing sectional curvature) which do not admit any spin or spin$${}^c$$-structure. But since the existence of spin, resp. of spin$${}^c$$-structures is detected by the second Stiefel-Whitney class, I am strongly led to believe that there is no way of defining these à la Chern-Weil. The reason is that the phrase 'à la Chern-Weil' entails for me that you use in an essential way the curvature tensor, but in these examples it vanishes.